Schreiber integration of ∞-Lie algebroid valued differential forms

An integration of ∞-Lie algebroid valued differential forms $\omega : \Pi^{inf}(X) \to \mathfrak{a}$ is an extension $\int \omega$ from the infinitesimal path ∞-groupoid? $\Pi^{inf}(X)$ to the finite path ∞-groupoid $\Pi(X)$

$$\array{ \Pi^{inf}(X) &\stackrel{\omega}{\to}& \mathfrak{a} \\ \downarrow && \downarrow \\ \Pi(x) &\stackrel{\int \omega}{\to}& A }$$

after injecting the ∞-Lie algebroid $\mathfrak{a}$ into one of its corresponding ∞-Lie groupoids $A$.

Idea

Assume a context given by a smooth (∞,1)-topos with its notion of ∞-Lie theory.

For $A$ some ∞-Lie groupoid and $U \in C$ a test domain, a morphism $\phi : \Pi^{inf}(U) \to A$ from the infinitesimal path ∞-groupoid? factors – by definition of ∞-Lie algebroid – through the ∞-Lie algebroid $\mathfrak{a}$ of $A$

$$\Pi^{inf}(U) \stackrel{\omega}{\to} \mathfrak{a} \hookrightarrow A$$

The first morphism here constitutes the set of ∞-Lie algebroid valued differential forms on $U$ that is encoded by $\phi$.

We want to specify what it means to integrate these forms.

Whereas $\omega$ colors infinitesimal intervals on $U$ by infinitesimal morphisms in $A$, its integration should be a rule that labels finite intervals by finite morphisms, such that the restriction to any infinitesimal interval within a finite interval reproduces the assignment of $\omgea$.

Formally, this desideratum clearly means that integration $\int \omega$ of $\omega$ is an extension

$$\array{ \Pi^{inf}(U) &\stackrel{\omega}{\to}& A \\ \downarrow & \nearrow_{\int \omega} \\ \Pi(U) }$$

of $\omega$ through the inclusion $\Pi^{inf}(U) \hookrightarrow \Pi(U)$ of the infinitesimal path ∞-groupoid? in the path ∞-groupoid.

More generally, we can ask for a relative integration : we may have a situation where a coloring of finite intervals

$$\Pi(U) \to B$$

is already given, but infinitesimally there is also a lift specified of this through some map $A \to B$, i.e. a diagram

$$\array{ \Pi^{inf}(U) &\stackrel{\omega}{\to}& A \\ \downarrow && \downarrow \\ \Pi(U) &\to& B }$$

Integration should exist when the map $A \to B$ in principle admits such a lift and should be given by that lift. It should be unique up to equivalence.

Details

We discuss a case in which integration of $\infty$-Lie algebroid valued forms always exsist.

Let $\mathfrak{a}$ be an ∞-Lie algebroid.

Write for short $\mathfrak{a}_# := \mathfrak{a}_{#^{inf}}$ in the following for the image of $\mathfrak{a}$ under the right adjoint of the infinitesimal path ∞-groupoid? functor $\Pi^{inf}$. And write $\mathfrak{a}_{# 0}$ for the corresponding presheaf of 0-cells.

By the reasoning an ∞-Lie differentiation and integration we have

• the $\infty$-Lie algebroid of $\mathfrak{a}$ is $\mathfrak{a}$ itself:

  $$Lie(\mathfrak{a}) := \Pi^{inf}(\mathfrak{a}_{# 0}) = \mathfrak{a} \,.$$

• the ∞-Lie groupoid integrating $\mathfrak{a}$ is

  $$A = \Pi(\mathfrak{a}_{# 0}) \,.$$

For $X$ simplicially discrete, every morphism

$$\omega : \Pi^{inf}(X) \to \mathfrak{a}$$

corresponding by adjunction to a morphism

$$\tilde \omega : X \to \mathfrak{a}_{#_0}$$

induces a morphism

$$\int \omega := \Pi(\tilde \omega) : \Pi(X) \to \Pi(\mathfrak{a}_{# 0}) = A$$

that fits into the diagram

$$\array{ \Pi^{inf}(X) &\stackrel{\omega}{\to}& \Pi^{inf}(\mathfrak{a}_{# 0}) &=& \mathfrak{a} \\ \downarrow && \downarrow && \downarrow \\ \Pi(X) &\stackrel{\int \omega}{\to}& \Pi(\mathfrak{a}_{# 0}) & = & A } \,.$$